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Theorem nfrdg 7397
 Description: Bound-variable hypothesis builder for the recursive definition generator. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
Hypotheses
Ref Expression
nfrdg.1 𝑥𝐹
nfrdg.2 𝑥𝐴
Assertion
Ref Expression
nfrdg 𝑥rec(𝐹, 𝐴)

Proof of Theorem nfrdg
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 df-rdg 7393 . 2 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
2 nfcv 2751 . . . 4 𝑥V
3 nfv 1830 . . . . 5 𝑥 𝑔 = ∅
4 nfrdg.2 . . . . 5 𝑥𝐴
5 nfv 1830 . . . . . 6 𝑥Lim dom 𝑔
6 nfcv 2751 . . . . . 6 𝑥 ran 𝑔
7 nfrdg.1 . . . . . . 7 𝑥𝐹
8 nfcv 2751 . . . . . . 7 𝑥(𝑔 dom 𝑔)
97, 8nffv 6110 . . . . . 6 𝑥(𝐹‘(𝑔 dom 𝑔))
105, 6, 9nfif 4065 . . . . 5 𝑥if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
113, 4, 10nfif 4065 . . . 4 𝑥if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
122, 11nfmpt 4674 . . 3 𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1312nfrecs 7358 . 2 𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
141, 13nfcxfr 2749 1 𝑥rec(𝐹, 𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  Ⅎwnfc 2738  Vcvv 3173  ∅c0 3874  ifcif 4036  ∪ cuni 4372   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  Lim wlim 5641  ‘cfv 5804  recscrecs 7354  reccrdg 7392 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fv 5812  df-wrecs 7294  df-recs 7355  df-rdg 7393 This theorem is referenced by:  rdgsucmptf  7411  rdgsucmptnf  7412  frsucmpt  7420  frsucmptn  7421  nfseq  12673  trpredlem1  30971  trpredrec  30982  finxpreclem6  32409
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